math lessons - Euler's equations

This page discusses rigid body dynamics. For compressible fluid flow, see Euler equations.

In physics, Euler's equations govern the rotation of a rigid body. We choose the body fixed axes to be principal axes of inertia. This will make the calculations easier, since we can now split the change in angular momentum into a component that describes the change of the size of $\mathbf{L}$ and another component that compensates for the change in direction of $\mathbf{L}$ .

The equotations are:

$\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}+\mathbf{\omega}\times\mathbf{L}=\frac{d\mathbf{L}}{dt}=\mathbf{N}$

where $\mathbf{L}$ is the projection of the angular momentum in the body fixed axes, $\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}$ the change of the angular momentum of the body with respect to the body fixed axes, $\mathbf{\omega}$ the rate of change of the Euler angles of the body connected axes with respect to the space axes, and $\mathbf{N}$ the external torque.

Proof

If we replace $\mathbf{L}$ with its components $I_1\omega_1\mathbf{e}_1 + I_2\omega_2\mathbf{e}_2 + I_3\omega_3\mathbf{e}_3$ we can replace $\frac{d\mathbf{L}}{dt}$ with $I_1\dot{\omega}_1\mathbf{e}_1 + I_2\dot{\omega}_2\mathbf{e}_2+I_3\dot{\omega}_3\mathbf{e}_3 + \frac{d\mathbf{e}_1}{dt}\omega_1I_1 + \frac{d\mathbf{e}_2}{dt}\omega_2I_2 + \frac{d\mathbf{e}_3}{dt}\omega_3I_3$ . If we choose the base vectors $(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)$ to be the body fixed axes, the first three terms are equal to $\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}$ and the rest is $\mathbf{\omega}\times\mathbf{L}$

Application

In component form, the Euler equations become

$\begin{matrix} N_1 &=& I_1\dot{\omega}_1+(I_3-I_2)\omega_2\omega_3\\ N_2 &=& I_2\dot{\omega}_2+(I_1-I_3)\omega_3\omega_1\\ N_3 &=& I_3\dot{\omega}_3+(I_2-I_1)\omega_1\omega_2\\ \end{matrix}$

It is also possible to use these equotations if the axes in which $\left(\frac{d\mathbf{L}}{dt}\right)_\mathrm{relative}$ is described are not connected to the body. $\mathbf{\omega}$ should then be replaced with the rotation of the axes instead of the rotation of the body. It is however still required that the chosen axes are still principal axes of intertia! This form of the Euler equotations is handy for rotation symmetric objects that allow some of the principle axes of rotation to be chosen freely.

See Poinsot's construction .

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Euler's equations

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Application

Mathematics Encyclopedia and Lessons

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Euler's equations

Proof

Application

Popular
Subjects